Observed-data likelihood ($L_{\mathrm{obs}}(\theta )=\int f_{\theta }(y_{\mathrm{obs}},y_{\mathrm{mis}})d{y}_{\mathrm{mis}}$)

An observed-data likelihood integrates the full-data statistical probability density over unobserved values, rather than inserting arbitrary values or discarding incomplete records. When the observation pattern is random, its mechanism belongs to the joint observation model unless the conditions for an ignorable missingness mechanism permit its factor to be omitted.